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(54                        ANALYTICAL MECHANICS
In order to express equation (4) in terms of rectangular coordinates we replace tan 0 by (f[ and oh turn
But ds2=dx2+dy2, therefore eliminating dx between this equation and equation (5) and separating the variables
and then integrating
where a =  and c is the constant of integration.
Let the o>axis be so chosen that when .s -. o, //    , then c = 0.   Therefore
j/ = V$H-a"2",   or *s* ~ \ //-  n\                    (7)
Differentiating equation (7), squaring and replacing (/- by we have
Solving for <fa,
_ __..... a%
where i - V-1.   Integrating equation (8) we get
ix            t/